This problem involves expressing the integral $\int_1^2 \frac{x}{1 + x^5} \, dx$ as the limit of a Riemann sum. Let's solve it step by step:

Step 1: Identify the general form of the Riemann sum

For an integral of the form $\int_a^b f(x) \, dx$, the corresponding Riemann sum is: $\lim_{n \to \infty} \sum_{i=1}^n f(x_i^*) \Delta x,$ where:

• $\Delta x = \frac{b - a}{n}$, the width of each subinterval,
• $x_i^*$, a sample point in the $i$-th subinterval.

Step 2: Parameters for this integral

Here:

• $a = 1$, $b = 2$,
• $f(x) = \frac{x}{1 + x^5}$,
• $\Delta x = \frac{2 - 1}{n} = \frac{1}{n}$,
• $x_i^*$ can be chosen as the right endpoint of the subinterval: $x_i = 1 + \frac{i}{n}$.

Step 3: Construct the Riemann sum

The sum becomes: $\sum_{i=1}^n \frac{x_i}{1 + x_i^5} \Delta x,$ where $x_i = 1 + \frac{i}{n}$ and $\Delta x = \frac{1}{n}$. Substituting $x_i$ and $\Delta x$, the sum becomes: $\sum_{i=1}^n \frac{1 + \frac{i}{n}}{1 + \left(1 + \frac{i}{n}\right)^5} \cdot \frac{1}{n}.$

Step 4: Write the limit of the sum

As $n \to \infty$, the integral is: $\int_1^2 \frac{x}{1 + x^5} \, dx = \lim_{n \to \infty} \sum_{i=1}^n \frac{1 + \frac{i}{n}}{1 + \left(1 + \frac{i}{n}\right)^5} \cdot \frac{1}{n}.$

Step 5: Match with the choices

The correct expression in the choices is C: $\lim_{n \to \infty} \sum_{i=1}^n \frac{\left(1 + \frac{i}{n}\right)}{1 + \left(1 + \frac{i}{n}\right)^5} \cdot \frac{1}{n}.$

Let me know if you would like further explanation or details.

Here are 5 related questions to deepen understanding:

1. How does choosing different sample points ($x_i^*$) affect the Riemann sum?
2. What happens to the accuracy of the Riemann sum as $n$ increases?
3. How would the integral change if the limits of integration were reversed?
4. Can you derive the Riemann sum for a different function, such as $\int_0^1 e^x \, dx$?
5. What is the geometric interpretation of a Riemann sum?

Tip: Practice visualizing Riemann sums as rectangles under a curve to develop intuition about integration!